Just as time was separated from the wave equation, time can also be separated directly from the fields in Maxwell's equations.  The electric and magnetic fields with the time variation separated is defined as,
\begin{align}
	\vec{\mathcal{E}}(\vec{r},t) &= \vec{E}(\vec{r})T(t)\label{eqn:sepeqnrtE}\\
	\vec{\mathcal{H}}(\vec{r},t) &= \vec{H}(\vec{r})T(t)\label{eqn:sepeqnrtH}
\end{align}
where the source terms are also separated as,
\begin{align}
\vec{\mathcal{J}}_i(\vec{r},t)&=\vec{J}_i(\vec{r})T(t)\label{eqn:sepJi}\\
\vec{\mathcal{M}}_i(\vec{r},t)&=\vec{M}_i(\vec{r})T(t)\label{eqn:sepMi}\\
\rho_{ev}(\vec{r},t)&=q_{ev}(\vec{r})T(t)\label{eqn:sepqev}\\
\rho_{mv}(\vec{r},t)&=q_{mv}(\vec{r})T(t)\label{eqn:sepqmv}
\end{align}
Substituting (\ref{eqn:sepeqnrtE})--(\ref{eqn:sepqmv}) into (\ref{eqn:geflsf})--(\ref{eqn:alsf}) yields,
\begin{align}
    \nabla\cdot\vec{E}T&=\frac{q_{ev}}{\varepsilon}T\label{eqn:geflsfsep}\\
    \nabla\cdot\vec{H}T&=\frac{q_{mv}}{\mu}T\label{eqn:gmflsfsep}\\
    \nabla\times\;\vec{E}T&=-\mu\frac{\partial\vec{H}T}{\partial{t}}-\sigma_m\vec{H}T-\vec{M}_iT&&\label{eqn:flsfsep}\\
    \nabla\times\,\vec{H}T&=\varepsilon\frac{\partial\vec{E}T}{\partial{t}}+\sigma_e\vec{E}T+\vec{J}_iT\label{eqn:alsfsep}
\end{align}
Rearranging and dividing both sides by $T(t)$ yields,
\begin{align}
    \nabla\cdot\vec{E}&=\frac{q_{ev}}{\varepsilon}\label{eqn:geflsfsep2}\\
    \nabla\cdot\vec{H}&=\frac{q_{mv}}{\mu}\label{eqn:gmflsfsep2}\\
    \nabla\times\;\vec{E}&=-(\mu{T'/T}+\sigma_m)\vec{H}-\vec{M}_i\label{eqn:flsfsep2}\\
    \nabla\times\,\vec{H}&=(\varepsilon{T'/T}+\sigma_e)\vec{E}+\vec{J}_i\label{eqn:alsfsep2}
\end{align}
If sinusoidal temporal variations are assumed then by substituting (\ref{eqn:steadystate}) and (\ref{eqn:dsteadystate}) into (\ref{eqn:geflsfsep2})--(\ref{eqn:alsfsep2}) yields,
\begin{align}
    \nabla\cdot\varepsilon\vec{E}&=q_{ev}\label{eqn:geflsfsep3}\\
    \nabla\cdot\mu\vec{H}&=q_{mv}\label{eqn:gmflsfsep3}\\
    \nabla\times\;\vec{E}&=-j\omega\mu'\vec{H}-\vec{M}_i\label{eqn:flsfsep3}\\
    \nabla\times\,\vec{H}&=j\omega\varepsilon'\vec{E}+\vec{J}_i\label{eqn:alsfsep3}
\end{align}
where $\sigma_m'$ and $\sigma_e'$ are defined in (\ref{eqn:sigma_m_primed}) and (\ref{eqn:sigma_e_primed}) and where $\mu'$ and $\varepsilon'$ are defined in (\ref{eqn:mu_complex}) and (\ref{eqn:eps_complex}).